Diffusions on Finsler Manifolds

Abstract

Let M be a finite-dimensional Finsler manifold with arc element ds = L(x ¹,… , x ⁿ, dx ¹,… , dx ⁿ), L being a non-negative smooth homogeneous function of degree one in dx ⁱ. The Finsler metric tensor, where will be assumed to be positive definite for all x ∈ M and 0 ≠y ∈TM _x. (For some examples, y ¹…y ⁿ ≠ 0, is required.) Let M be equipped with the Cartan connection being the nonlinear, the horizontal, and the vertical connection coefficients, which can be expressed as where as above, with and. Here and throughout the paper we use the standard summation convention. The horizontal and vertical covariant derivatives of a Finsler vector field A ⁱ(x, y) will be denoted by These formulae can be extended to any Finsler tensor fields in the usual way. The Cartan connection is metrical, i.e., (1.1)

Partially supported by NSERC A-7667. This article appeared in Rep. Math. Phys., 33, (1993), 303-315. and deflection-free, i.e.,. A systematic presentation of Finsler geometry can be found in the classical monograph [13] by Rund. For a modern approach to Finsler manifolds see Matsumoto’s monograph [11]. We also refer to Cartan’s original work [3].